Experimental realisation of multi-qubit gates using electron paramagnetic resonance

Quantum information processing promises to revolutionise computing; quantum algorithms have been discovered that address common tasks significantly more efficiently than their classical counterparts. For a physical system to be a viable quantum computer it must be possible to initialise its quantum state, to realise a set of universal quantum logic gates, including at least one multi-qubit gate, and to make measurements of qubit states. Molecular Electron Spin Qubits (MESQs) have been proposed to fulfil these criteria, as their bottom-up synthesis should facilitate tuning properties as desired and the reproducible production of multi-MESQ structures. Here we explore how to perform a two-qubit entangling gate on a multi-MESQ system, and how to readout the state via quantum state tomography. We propose methods of accomplishing both procedures using multifrequency pulse Electron Paramagnetic Resonance (EPR) and apply them to a model MESQ structure consisting of two nitroxide spin centres. Our results confirm the methodological principles and shed light on the experimental hurdles which must be overcome to realise a demonstration of controlled entanglement on this system.


Channel
Phase where () is the form factor, () is the background function, and  is the acquisition offset. 2 The acquisition offset is caused by imperfect calibration of the detection voltages and is generally cancelled out by a phase cycle.In the absence of phase cycling this represents a constant proportional to the number of scans.The background function is caused by low frequency inter-molecular interspin interactions, which give rise to low frequency oscillations in the time trace and in homogeneous samples can be modelled as an exponential decay, () =  )*+ . 3The decay rate  depends on (amongst other things) the spin concentration in the sample and the excitation fraction of the pump pulse.The form factor contains all intra-molecular information, and can be expressed as: where  " is the unmodulated echo intensity,  " is the amplitude of the modulations,  , () is the cosine modulation function,  -() is the sine modulation function,  is the depth of the cosine modulation, and  is the depth of the sine modulation.The modulation functions can be given as: where  is the subpopulation of molecules in the sample in which one spin is detected and the other is inverted by the pump pulse, and  ./ 0is the coupling frequency of the spin pair.If the anisotropy of the spin systems can be neglected ( 1 ≈  2 ≈  3 for both spins) and the high-field limit applies (| .−  / | ≪  ./ ) then the inter-spin interaction is given by: where  " is the vacuum permeability,  / is the Bohr magneton, ℏ is the reduced Planck constant,  .0 and  / 0 are the -values of the two spins,  ./ 0is the inter-spin distance,  ./ 0is the angle between the inter-spin vector and the external magnetic field, and  ./0 is the super-exchange (through-bond) interaction between the two spins. 4e remaining parameters in the form factor relate to the state of the system before the application of the detection sequence.The unmodulated echo intensity is  " = < > 45 ?6 , where  > 45 is the relevant in-phase spin operator for the experiment, the subscript d denotes the expectation value over the sub-population of spins resonant with the detection frequency, and  is a proportionality constant.Similarly, the modulation depths can be expressed  = < > 45 ?6∧8 and  = < > '594 ?6∧8 , where  > '594 is the relevant anti-phase operator, the subscript d ∧ p denotes the expectation value is taken over the sub-population of spins resonant with the detection frequency whose partner spins are inverted by the pump pulse, and  is a proportionality constant smaller than  as it accounts for partial inversion of partner spins.

Supplementary Note 2.2 Pseudo-fidelity
State fidelity metrics aim to provide a numerical value for the similarity between two quantum states, such as ℱ( F $ ,  F : ) = Htr KLM F $  F : M F $ NO : . 5 For magnetic resonance measurements in the high temperature limit, the both states will be dominated by the identity element, and so their fidelity by this metric will be approximately equal to one.A more meaningful metric must compare their deviation density matrices, which we accomplish by treating the expectation values of the nonidentity Cartesian product operators as vectors in the generalised Bloch sphere.In this representation the similarity between two states can be thought of as the angle between them, and we use the cosine of this angle to define the pseudo-fidelity: where  is the Hilbert dimension, and: where  ⃗ is the vector representation of  F, and  and  are the expectation values of the basis operators in  F $ and  F : , respectively (i.e. the elements of  ⃗ $ and  ⃗ : ).Note that this is not a generally acceptable metric as it does not fulfil many requirements of true fidelity metrics, however, ℱ P ( F $ ,  F $ ) = 1, ℱ P ( F $ ,  F : ) = ℱ P ( F : ,  F $ ), and ℱ P ( F $ ,  F : ) = ℱ P ( F $  < ,  F :  < ) where  is a unitary operation.Further, if  F $ and  F : are both pure ℱ P ( F $ ,  F : ) = ℱ( F $ ,  F : ).If the absolute magnitude of the observables cannot be determined, it is common to normalise the vectors, resulting in a normalised pseudo-fidelity: where: It should be noted that the fitted experimental state was normalised by the magnitude of the calculated expectation values because the experimental starting state is unknown.This means that the measurement is insensitive to an overall loss in magnetisation, and therefore this measurement of this pseudo-fidelity is a normalised pseudo-fidelity and an upper-bound (see further discussion later in the 'Relation to experiment' section in the main text).
overlapping in time.The resulting fitted coupled density matrix was then rotated about  to maximise its pseudo-fidelity to a reference density matrix of an ideal entangling gate,  F (HA =  j 1  j 3 +  j 3  j 1 , accounting for errors in pulse and detection phases. The fit procedure was repeated for a range of filter pulse flip probabilities, ranging from 0.01 to 1.00 in steps of 0.01.While the highest pseudo-fidelity of 0.790 was found when  AB48 = 0.45, while the lowest cost occurred at  AB48 = 0.42, for which the pseudo-fidelity was marginally lower at 0.789 (Supplementary Figure 6).All fits shown correspond to the lowest cost solution.Supplementary Supplementary Figure 8: Experimental DEERATom time traces (black) and corresponding fits (blue and orange) recorded with the pairs of operations a. X " 1 # detected at  " , b. X " 1 # detected at  # , c. X " X # detected at  " , d. X " X # detected at  # , e. X " Y # detected at  " , and f.X " Y # detected at  # .All data have been background corrected and normalised, with the real component denoted as I and the imaginary part as Q.For illustrative purposes full product operator calculations for  j 2 and 2 j 1  j 3 are shown for a pumprefocus building block detecting on spin S. The results of the calculations for all operators detecting at both frequencies are shown in Supplementary

Figures 7 , 7 :
Figures 7, 8 and 9 show the normalised, background corrected time traces and corresponding fits for all pairs of operations.

9 :
Experimental DEERATom time traces (black) and corresponding fits (blue and orange) recorded with the pairs of operations a. Y " 1 # detected at  " , b. Y " 1 # detected at  # , c. Y " X # detected at  " , d. Y " X # detected at  # , e. Y " Y # detected at  " , and f.Y " Y # detected at  # .All data have been background corrected and normalised, with the real component denoted as I and the imaginary part as Q.